In an (∞,1)-category the idea is the same, except that the notion of idempotent is more complicated. Instead of just requiring that e∘e=ee\circ e = e, we need an equivalencee∘e≃ee\circ e \simeq e, together with higher coherence data saying that, for instance, the two derived equivalences e∘e∘e≃ee\circ e\circ e \simeq e are equivalent, and so on up. In particular, being idempotent is no longer a property of a morphism, but structure on it.

It is still true that a splitting of an idempotent in an (∞,1)(\infty,1)-category is a limit or colimit of that idempotent (now regarded as a diagram with all its higher coherence data), but this limit is no longer a finite limit; thus an (∞,1)(\infty,1)-category can have all finite limits without being idempotent-complete.

Definition

Definition

Let Idem+Idem^+ be the nerve of the free 1-category containing aretraction, with e:X→Xe:X\to X the idempotent, r:X→Yr:X\to Y the retraction, and i:Y→Xi:Y\to X the section (and e=ire = i r and ri=1Yr i = 1_Y). Let IdemIdem be the similar nerve of the free 1-category containing an idempotent, which is the full sub-simplicial set of Idem+Idem^+ spanned by the object XX. Let RetRet be the image in Idem+Idem^+ of the 2-simplex Δ2→Idem+\Delta^2 \to Idem^+ exhibiting the composite ri=1Yr i = 1_Y; thus RetRet is also the quotient of Δ2\Delta^2 that collapses the 1-face to a point.

Coherent vs incoherent idempotents

We may also ask how idempotent-completeness of CC is related to that of its homotopy categoryhCh C. An idempotent in hCh C is an “incoherent idempotent” in CC, i.e. a map e:X→Xe:X\to X such that e∼e2e \sim e^2, but without any higher coherence conditions. In this case we have:

Theorem

(HA Lemma 1.2.4.6) If CC is stable, then CC is idempotent-complete (i.e. every coherent idempotent is effective) if and only if hCh C is (as a 1-category).

However, if CC is not stable, this is false. The following counterexample in ?Gpd? is constructed in Warning 1.2.4.8 of HA. Let λ:G→G\lambda : G \to G be an injective but non-bijective group homomorphism such that λ\lambda and λ2\lambda^2 are conjugate. (One such is obtained by letting GG be the group of endpoint-fixing homeomorphisms of [0,1][0,1], with λ(g)\lambda(g) acting as a scaled version of gg on [0,12][0,\frac 1 2] and the identity on [12,1][\frac 1 2,1]. Then λ(g)∘h=h∘λ2(g)\lambda(g) \circ h = h \circ \lambda^2(g) for any hh such that h(t)=2th(t) = 2t for t∈[0,14]t \in [0,\frac 1 4].)

Then Bλ:BG→BGB\lambda : B G \to B G is homotopic to Bλ2B\lambda^2, hence idempotent in the homotopy category. If it could be lifted to a coherent idempotent, then the colimit of the diagram

would be its splitting, and hence the map BG→colim(BG→BλBG→⋯)B G \to \colim (B G\xrightarrow{B \lambda} B G \to\cdots) would have a section. Passing to fundamental groups, G→colim(G→λG→⋯)G \to \colim (G\xrightarrow{\lambda} G \to\cdots) would also have a section; but this is impossible as λ\lambda is injective but not surjective.

However, we do have the following:

Theorem

(HA Lemma 7.3.5.14) A morphism ee in an (∞,1)(\infty,1)-category CC is idempotent (i.e. e:Δ1→Ce:\Delta^1 \to C extends to IdemIdem) if and only if there is a homotopy h:e∼e2h : e \sim e^2 such that h∘1∼1∘hh\circ 1 \sim 1\circ h.

In other words, an incoherent idempotent can be fully coherentified as soon as it admits one additional coherence datum.